One of the drawbacks of the type II progressive censoring scheme is that the length of the experiment can be very large. Because of that, recently a new censoring scheme named as the type II progressively hybrid censored scheme has received considerable interest among the statisticians. In this paper, the statistical inference for the half-logistic distribution is discussed based on the progressively type II hybrid censored samples. The maximum likelihood estimator, the approximate maximum likelihood estimator and the Bayes estimator of parameter using Lindley approximation and MCMC method are obtained. Asymptotic confidence intervals, Bootstrap confidence intervals and Bayesian credible intervals are obtained. Different point and interval estimators are compared using Monte Carlo simulation. A real data set is presented for illustrative purposes.

In this paper, the smallest confidence region is obtained for the location and scale parameters of the two-parameter exponential distribution. For this purpose, we use constrained optimization problems. We first provide some suitable pivotal quantities to obtain a balanced confidence region. We then obtain the smallest confidence region by minimizing the area of the confidence region using the Lagrangian method. Two numerical examples are presented to illustrate the proposed methods. Finally, some applications of proposed joint confidence regions in hypothesis testing and the construction of confidence bands are discussed.